A parametrization of a circle by arc length may be written as

$\gamma (t)=c+r\mathrm{cos}(t/r){e}_{1}+r\mathrm{sin}(t/r){e}_{2}.$

Suppose $\beta $ is an unit speed curve such that its curvature $\kappa $ satisfies $\kappa (0)>0$.

How to show there is one, and only one, circle which approximates $\beta $ in near $t=0$ in the sense

$\gamma (0)=\beta (0),{\gamma}^{{}^{\prime}}(0)={\beta}^{{}^{\prime}}(0)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\gamma}^{{}^{\u2033}}(0)={\beta}^{{}^{\u2033}}(0).$

I suppose we must use Taylor's formula, but I wasn't able to solve this problem.